Math 575 – Review Materials - Exam 1
I. You should be able to recognize, define, state and give examples of the following
concepts. (Several of these will be on the exam)
K n , Cn , Pn , K n,m
boundary
center of a graph
complete graph
connected graph
cycle
diameter
end-vertex
good coloring
isolated vertex
Kuratowski’s Theorem
neighborhood of a vertex
Petersen Graph
Prüffer code
regular graph
spanning tree
tree
clique number ! (G)
bipartite graph
bridge
chromatic number
complete bipartite graph
cubic graph
degree of a vertex
distance between vertices
Euler’s Theorem
independent set of vertices
isomorphic graphs
length of a path
outerplanar graph
planar graph
radius
self-complementary graph
subdivision
color classes of a coloring
independence number ! (G)
bipartition
Cayley’s Theorem
complement of a graph
component
cut-vertex
degree sequence of a graph
eccentricity
face/region
induced subgraph
isomorphism
Möbius Strip
path
plane graph
Reconstruction Conjecture
spanning subgraph
subgraph
walk
Grötzsch Graph
II.
You should be able to handle problems similar to those on the homework problem
sets 1- 6, the quizzes, examples from class, and the review problems below.
III.
Know the statements of Kuratowski’s Theorem, Euler’s Theorem relating regions,
edges and vertices, and Cayley’s Theorem on the number of distinct trees on n
vertices – and be able to apply them.
IV.
Be Able to Prove:
•
If G has m edges, then 2m =
"
deg(v) .
v!V (G )
•
•
•
•
•
•
•
Euler's Theorem
The 5-color Theorem
Every tree on n vertices has n–1 edges
Every tree on n ≥ 2 vertices has at least two end vertices.
If G is a planar graph, then G contains a vertex of degree at most 5.
If r and d are the radius and diameter of a connected graph G, then r ≤ d ≤ 2r.
If G has no cycles and n–1 edges, then G is a tree.
V.
Some Additional Problems
1. Find a tree having the Prüffer code (1, 7, 9, 2, 1, 2, 9)
Answer:
2. How many distinct trees on the vertex set V = {1, 2, 3, 4, …, 12}, have exactly three
end-vertices?
!12$
!10$
Solution: # & ' 9 ' # & ' 8!
" 3%
" 2%
First choose three of the integers from 1 – 12 to be the end-vertices. These three integers
will not appear in the Prüffer code. Now each of the remaining 9 integers must be used at
least once in the Prüffer code, and so the 10-slots of the Prüffer code will have one
integer repeated twice and all the others used exactly once – for example if 10, 11, and 12
are the end-vertices, then the resulting Prüffer code might be (1, 4, 2, 6, 7, 9, 4, 3, 5, 8).
Where the integer 4 is repeated twice. So there are 9 ways to choose the integer that gets
repeated twice and then choose 2 of the 10 slots for the repeated integer and then arrange
the remaining 8 integers in 8! ways.
3. The degree sequence of a graph is simply a list of the degrees of the vertices in
ascending order.
If the degree sequence of G is 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 4, then what is the degree
sequence of the complement of G?
Answer: G has 11 vertices, and hence each vertex v of G satisfies,
deg G (v) = 10 ! deg G (v). . So the degree sequence of G is 6, 6, 7, 7, 8, 8, 8, 8, 8, 9, 9.
4. Which pairs of the connected graphs below are isomorphic? Verify your answer.
Solution: All four graphs are isomorphic.
5. Show that the graphs below are isomorphic by constructing a specific isomorphism.
s
f
r
w
t
z
x
y
G
e
g
a
b
d
c
H
Answer: An isomorphism is x ! a, y ! d, z ! e, w ! f , r ! g, s ! c, t ! b .
6. Suppose that G is a graph that contains 11 vertices and 31 edges and every vertex of
G has degree 4 or 7. How many vertices of G have degree 7?
Solution: G has 6 vertices of degree 7.
7. If A is a set of vertices of the graph G, then recall that the subgraph H, induced by A,
is the subgraph whose vertex set is V (H ) = A and two vertices are adjacent in H if
and only if they are adjacent in G.
1
Show that the number of triangles in a graph G is given by T (G) = " m(N(v))
3 v!V (G )
where m(A) denotes the number of edges in the subgraph induced by A.
Answer: Every edge in the neighborhood of a vertex v is the side of a triangle that
contains v. Consequently, every triangle gets counted in each m(N(v)) for each
vertex v that is in the triangle. Hence each triangle is counted exactly three times in
the given summation.
8. Prove: If G is regular of degree 4, then G does not contain a bridge.
Proof: Suppose that e = ab is a bridge of G. Then G ! e contains exactly two
components A and B with a ! A and b ! B . But then in the subgraph induced by A,
every vertex has degree 4 except for a which has degree 3. This is a contradiction,
since a graph must contain an even number of vertices of odd degree.
9. How many distinct trees on the vertex set {1, 2, 3, 4, 5, 6, 7, 8, 9} are there that are
isomorphic to each of the trees below?
! 9$
# & 5!
! 8$ ! 6$
" 4%
Answer:
for the left graph and 9# & # & for the graph on the right.
" 2% " 3%
2
10. How many distinct spanning trees are there in the graph below?
Answer: (3 ! 2 ! 3 + 8 + 5 ! 3 + 5 ! 3)(6 + 3 ! 3) = 56 ! 15 = 840
11. Suppose that G is a tree whose complement is a maximal planar graph.
(a). How many vertices does G have? Verify your answer.
(b). Can you find an actual example of such a tree?
Answer: (a). Suppose that G has n vertices. Then since G is a tree, G has n ! 1
! n$
(n ' 1)(n ' 2)
edges. So the complement of G has # & ' (n ' 1) =
edges. Also, since
" 2%
2
(n ! 1)(n ! 2)
the complement is maximal planar,
= 3n ! 6 . Solving this gives
2
n = 2 or n = 7 . The value n = 2 does not work. However, there is a tree on 7 vertices
that works – namely,
12. If a tree is picked at random on the vertex set {1,2,3,…,n} (n ≥ 4), what is the
probability pn that all of vertices 1, 2 and 3 are end-vertices? What is the limiting
value of pn ?
(n ! 3)n ! 2 " n ! 3 %n ! 2
1
=$
Answer: pn =
and so , lim pn = 3 .
n !2
'
n!"
n
# n &
e
13. How many distinct trees on the vertex set {1, 2, 3, 4, 5, 6, 7, 8} are there
in which deg(1) = 3 and deg(4) = 2 ?
! 6$
3
Answer: # & ' 4 ' 6
" 2%
14. Let G be a plane graph having all vertices of degree 4 or 5. Suppose too that G has 21
regions. There are 3 4-regions (regions bounded by 4 edges) and 18 3-regions.
How many vertices of G have degree 4 and how many have degree 5?
Solution: Suppose there are a vertices of degree 4 and b of degree 5.
There are 33 edges and 14 vertices, and so a + b = 14 and 4a + 5b = 66.
Solving simultaneously we get a = 4 and b = 10.
15. Suppose that G is a bridgeless plane graph with 26 vertices and 20 faces.
Each of the 19 non-exterior regions is bounded by 4 edges. How many edges
are in the boundary of the exterior regions? Show your work
Solution: Let x denote the number of edges in the exterior face and let m denote the
number of edges of G. Then, 2m =
"
F a face
!F = 19 # 4 + x = 76 + x .
So, x = 2m ! 76 . But, by Euler’s Theorem, n – m + f = 2 we get 26 – m + 20 = 2 so,
m = 44. Thus, x = 2 ! 44 " 76 = 12 . So there are 12 edges in the boundary of the
exterior face.
16. Show that there are exactly two non- isomorphic 4-regular graphs of order 7.
Hint: What about the complements of such graphs?
Proof: There are clearly just two non-isomorphic graphs that are 2-regular of order 7
– namely the cycle on 7 vertices and the graph consisting of the disjoint union of a 4and 3-cycle.
17. How do you know the graph below has no odd cycles?
Solution: Its vertices can be colored red and blue so that no two adjacent vertices
have the same color.
18. Suppose that G is a graph having 16 vertices – each of degree either 3 or 5. If G has
34 edges, then how many vertices of G have degree 3?
19. Suppose that G is a connected graph with deg(v) ! 4 for every vertex v in G.
Show that G must contain a cycle of length at least 4.
20. Suppose that G is a graph with V(G) = { v1, v2, ... v7} and that the graphs
G – v1, G – v2, ... and G – v7 are isomorphic to the graphs below.
(a) How many edges does G have?
(c) How many triangles does G have?
Answer: G has
G is the graph
(b) What is the degree sequence of G?
(d) What is G?
50
= 10 edges. Its degree sequence is 2 2 2 3 3 4 4. It has 3 triangles.
5
21. Complete the drawing below of K 6 in the torus.
22. The minimal degree of a graph is ! (G) = min {deg(v) : v "V (G)} ; i.e., ! (G) is the
smallest of all the degrees in G. Suppose that G has 13 vertices and 32 edges.
Show that ! (G) ≤ 4.
23. The maximal degree of a graph G is !(G) = max {deg(v) : v "V (G)} .
(a). Show that for any graph G, ! (G) " #(G) + 1 .
(b). Give an example of a graph for which ! (G) = "(G) + 1.
Solution: (b). Any odd cycle on n ≥ 5 vertices.
24. (a). How many distinct spanning trees are there for a complete graph on n vertices?
n! 2
Solution: n since every tree on a given set of n vertices is a spanning tree of the
complete graph on that same set of vertices.
(b). How many distinct spanning trees are there for K 3,n ?
Answer: n 2 3n !1
25. Use Cayley's Theorem to show that the sequence d1 ≤ d2 ≤ d3 ≤ … ≤ dn of positive
n
integers is the degree sequence of some tree if and only if
!d
i
= 2(n " 1) .
i=1
n
Proof: Clearly, if G is a tree then its degree sequence satisfies ! di = 2(n " 1) .
i=1
On the other hand, suppose that the sequence d1 ≤ d2 ≤ d3 ≤ … ≤ dn satisfies
n
!d
i
= 2(n " 1) . Then form the tree that has d1 ! 1 1' s d2 ! 1 2's, …, dn ! 1 n's in its
i=1
n
Prüffer code (this works because
i
i= 1
proper length of a Prüffer code.
n
" (d ! 1) = " d ! n = 2(n ! 1) ! n = n ! 2 – the
i
i =1
26. Suppose that G is a planar graph that does not contain a vertex of degree less than 5.
Show that G contains at least 12 vertices.
Solution: Suppose that every vertex of G has degree 5 or more, then
5n ! " deg(v) = 2m ! 2 ( 3n # 6 ) = 6n # 12 . So, n ≥ 12.
27. Show that the complement of a path on 7 vertices is not a planar graph.
Solution: Note that the complement contains a subdivision of K 3, 3 .
28. The crossing number of a graph is the fewest possible edge crossings over all possible
ways to draw the graph in the plane. Show that the crossing number of K 6 is 3.
29. Show how to embed K 6 on a möbius strip.
Show how to embed K 3, 3 on a torus and on a möbius strip.
30. How many trees on V = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} have exactly 5 end-vertices?
Hint: The number of onto functions from a set of 8 elements onto a set of 5 elements
is 1050.